As seen from Fig.REF , In Rényi formalism, the ratios of the two temperatures (entropies) decreases (increases) with dimension \(d\) , acquiring its highest (lowest) value for \(d=5\)  (\(d=4\) ), as 1 (\( \frac{\log {\left(2 \right)}}{\log {\left(3 \right)}}\approx 0.6309\) ). This implies that in lower dimensional spacetime the collapse of thermal radiation to form a black-hole is relatively the easiest compared with higher dimensional spacetime. It is remarkable in this respect that in five dimensions once the possibility of a black hole phase exists, \(T_R(5)=T_{min}(5)\) , a Hawking-Page phase transition readily happens, \(T_{HP}(5)=T_{min}(5)\)  and \(S_{HP}(5)=S_{min}(5)\) . Such a behavior is exhibited also in \(GB\)  formalism but only at infinite dimensions. Furthermore, we have the limit as \( \lim _{d\rightarrow \infty }T_{min}/T_{HP}=0\)  (\( \lim _{d\rightarrow \infty }S_{min}/S_{HP}=\infty \) ) in contrast with the anti de-Sitter space in Ref.[1]}, [2]}. This dissimilarity stems from the fact that Rényi pressure \(P_R\)  depends on the horizon radius \(r_h\) , Eq.(REF ), While pressure in anti de-Sitter spacetime is independent of \(r_h\) , Eq.(REF ).
